document.write( "Question 1045034: Please help me find the solution & properties of ellipse and graph\r
\n" ); document.write( "\n" ); document.write( "4x^2=8y^2+8x-4y=8\r
\n" ); document.write( "\n" ); document.write( "Center:
\n" ); document.write( "Vertices:
\n" ); document.write( "Endpoints of minor axis:
\n" ); document.write( "Foci: \n" ); document.write( "
Algebra.Com's Answer #660459 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
It must be \"4x%5E2%2B8y%5E2%2B8x-4y=8\" , or something like that.
\n" ); document.write( "\"4x%5E2%2B8y%5E2%2B8x-4y=8\"--->\"4x%5E2%2B8x%2B8y%5E2-4y=8\"--->\"4%28x%5E2%2B2x%29%2B8%28y%5E2-%281%2F2%29%2Ay%29=8\"
\n" ); document.write( "When you look at that you realize that you almost have two squares:
\n" ); document.write( "\"4%28x%5E2%2B2x%2Bred%281%29%29=4%28x%2B1%29%5E2\" and \"8%28y%5E2-%281%2F2%29%2Ay%2Bgreen%281%2F16%29%29=8%28y%5E2-1%2F4%29%5E2\" .
\n" ); document.write( "So, you add \"4%2Ared%281%29%2B8%2Agreen%281%2F16%29=4%2B1%2F2\" to both sides of the equal sign of the original equation to get
\n" ); document.write( "\"4x%5E2%2B8x%2B4%2B8y%5E2-4y%2B1%2F2=8%2B4%2B1%2F2\"
\n" ); document.write( "\"4%28x%5E2%2B2x%2B1%29%2B8%28y%5E2-%281%2F2%29%2Ay%2B1%2F16%29=25%2F2\"
\n" ); document.write( "\"4%28x%2B1%29%5E2%2B8%28y-1%2F4%29%5E2=25%2F2\"
\n" ); document.write( "At that point you realize that
\n" ); document.write( "since \"x\" and \"y\" only appear once,
\n" ); document.write( "and as \"%28x%2B1%29%5E2\" and \"%28y-1%2F4%29%5E2\" ,
\n" ); document.write( "the curve represented by that equation has vertical and horizontal axes of symmetry, given by
\n" ); document.write( "\"x%2B1=0\" <---> \"x=-1\" and
\n" ); document.write( "\"y-1%2F4=0\" <---> \"y=1%2F4\" .
\n" ); document.write( "So, the major and minor axes must be along those lines.
\n" ); document.write( "That means that the center is the point with \"highlight%28system%28x=-1%2Cy=1%2F4%29%29\" .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You also realize that the ellipse cannot stray too far from that center.
\n" ); document.write( "The horizontal and vertical distance from the center to a point on the ellipse,
\n" ); document.write( "\"abs%28x%2B1%29\" and \"abs%28y-1%2F4%29\"
\n" ); document.write( "have maximum possible values,
\n" ); document.write( "because \"%28x%2B1%29%5E2\" and \"%28y-1%2F4%29%5E2\" have maximum possible values.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "No matter what value \"y\" takes,
\n" ); document.write( "since \"%28y-1%2F4%29%5E2%3E=0\" <---> \"8%28y-1%2F4%29%5E2%3E=0\" ,
\n" ); document.write( "\"4%28x%2B1%29%5E2=25%2F2-8%28y-1%2F4%29%5E2%3C=25%2F2\" ,
\n" ); document.write( "and \"4%28x%2B1%29%5E2%3C=25%2F2\" means \"red%28%28x%2B1%29%5E2%3C=25%2F8%29\" .
\n" ); document.write( "so there are horizontal ends to the ellipse.
\n" ); document.write( "Similarly, no matter what value \"x\" takes,
\n" ); document.write( "there are vertical ends to the ellipse
\n" ); document.write( "Since \"%28x%2B1%29%5E2%3E=0\" <---> \"4%28x%2B1%29%5E2%3E=0\" ,
\n" ); document.write( "\"8%28y-1%2F4%29%5E2=25%2F2-4%28x%2B1%29%5E2%3C=25%2F2\" ,
\n" ); document.write( "and \"8%28y-1%2F4%29%5E2%3C=25%2F2\" <---> \"green%28%28y-1%2F4%29%5E2%3C=25%2F16%29\" .
\n" ); document.write( "Since \"25%2F16%3C25%2F8\" ,
\n" ); document.write( "the ellipse stretches farther in the horizontal) x-direction.
\n" ); document.write( "That means that the horizontal \"y=1%2F4\" axis is the major axis.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The ends of the ellipse along the major axis are the vertices.
\n" ); document.write( "Their distance to the center is the semi-major axis, \"a\" , given by
\n" ); document.write( "\"red%28%28x%2B1%29%5E2%3C=25%2F8=a%5E2%29\" <--> \"abs%28x%2B1%29%5E2%3C=sqrt%2825%2F8%29=sqrt%2825%2A2%2F16%29=5sqrt%282%29%2F4\" .
\n" ); document.write( "So, the vertices have
\n" ); document.write( "\"x%2B1=-5sqrt%282%29%2F4\" <--> \"highlight%28x=-1-5sqrt%282%29%2F4%29\" <--> \"highlight%28x=%28-4-5sqrt%282%29%29%2F4%29\" and
\n" ); document.write( "\"x%2B1=5sqrt%282%29%2F4\" <--> \"highlight%28x=-1%2B5sqrt%282%29%2F4%29\" <--> \"highlight%28x=%28-4%2B5sqrt%282%29%29%2F4%29\" ,
\n" ); document.write( "along with \"highlight%28y=1%2F4%29\" .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The minor axis, along the line \"x=-1%29\" , has ends at a distance \"b\" from the center.
\n" ); document.write( "That distance is called the semi-minor axis and is given by
\n" ); document.write( "\"green%28%28y-1%2F4%29%5E2%3C=25%2F16=b%5E2%29\" <---> \"abs%28y-1%2F4%29%3C=sqrt%2825%2F16%29\" <---> \"abs%28y-1%2F4%29%3C=5%2F4\" .
\n" ); document.write( "So, the ends of the (vertical) minor axis of symmetry of the ellipse have
\n" ); document.write( "\"y=1%2F4-5%2F4=highlight%28-1%29\" and \"y=1%2F4%2B5%2F4=highlight%283%2F2%29\" ,
\n" ); document.write( "along with \"highlight%28x=-1%29\" .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The foci of the ellipse are along the major axis, \"y=1%2F4%29\" ,
\n" ); document.write( "at a distance \"c\" from the center of the ellipse,
\n" ); document.write( "and we know that \"a%5E2=b%5E2%2Bc%5E2\" .
\n" ); document.write( "Substituting the values found before,
\n" ); document.write( "\"red%2825%2F8=a%5E2%29\" and \"green%28%28y-1%2F4%29%5E2%3C=25%2F16=b%5E2%29\" , we get
\n" ); document.write( "\"25%2F8=25%2F16%2Bc%5E2\" ---> \"c%5E2=25%2F8-25%2F16\" ---> \"c%5E2=50%2F16-25%2F16\" ---> \"c%5E2=25%2F16\" ---> \"c=sqrt%2825%2F16%29\" ---> \"c=5%2F4\" .
\n" ); document.write( "So, the coordinates for the foci are \"highlight%28y=1%2F4%29\" , along with
\n" ); document.write( "\"x=-1-5%2F4=-4%2F4-5%2F4=highlight%28-9%2F4%29\" for one focus, and
\n" ); document.write( "\"x=-1%2B5%2F4=-4%2F4%2B5%2F4=highlight%281%2F4%29\" for the other focus. \n" ); document.write( "
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